Final answer:
To find the partial derivative of w with respect to t, we need to use the Chain Rule. We first find the partial derivatives of x, y, and z with respect to u and v. Then, we substitute these partial derivatives into the Chain Rule expression and simplify.
Step-by-step explanation:
The Chain Rule is a method for finding the derivative of a composition of two or more functions. In this case, we need to find the partial derivative of w with respect to t. Using the Chain Rule, we first find the partial derivatives of x, y, and z with respect to u and v.
x = sin(u - v), so ∂x/∂u = cos(u - v) and ∂x/∂v = -cos(u - v).
y = √(u + v), so ∂y/∂u = 1/ (2√(u + v)) and ∂y/∂v = 1/ (2√(u + v)).
z = u/v, so ∂z/∂u = 1/v and ∂z/∂v = -u/v^2.
Now, we can use the Chain Rule to find the partial derivative of w with respect to t:
∂w/∂t = ∂w/∂x * ∂x/∂u * ∂u/∂t + ∂w/∂x * ∂x/∂v * ∂v/∂t + ∂w/∂y * ∂y/∂u * ∂u/∂t + ∂w/∂y * ∂y/∂v * ∂v/∂t + ∂w/∂z * ∂z/∂u * ∂u/∂t + ∂w/∂z * ∂z/∂v * ∂v/∂t.
Substitute the partial derivatives we found earlier to evaluate the expression.